Question

Multivariable calculus. Line integral

Answer 1

Parameterize $C_R$ by

$\mathbf r(t)=(R\cos t,R\sin t)$

with $0\le t\le2\pi$.

Plug $\mathbf r$ into $\mathbf G$ and compute the differential $\mathrm d\mathbf r$:

$\mathbf G(\mathbf r(t))=\dfrac{(R\cos t,R\sin t)}{\sqrt{R^2-1}}$

$\mathrm d\mathbf r=(-R\sin t,R\cos t)\,\mathrm dt$

Plug everything into the integral and evaluate:

$\displaystyle\int_{C_R}\mathbf G\cdot\mathrm d\mathbf r=\int_0^{2\pi}0\,\mathrm dt=0$